arXiv subjects

Thomas Führer

Publications and source records attributed to Thomas Führer.

At least 37 records · Page 2Linked to original sources

arXiv2022

First-order system least-squares finite element method for singularly perturbed Darcy equations

We define and analyse a least-squares finite element method for a first-order reformulation of a scaled Brinkman model of fluid flow through porous media. We introduce a pseudostress variable that allows to eliminate the pressure variable from the system. It can be recovered by a simple post-processing. It is shown that the least-squares functional is uniformly equivalent, i.e., independent of the singular perturbation parameter, to a parameter dependent norm. This norm equivalence implies that the least-squares functional evaluated in the discrete solution provides an efficient and reliable a posteriori error estimator. Numerical experiments are presented.

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arXiv2022

Space-time finite element methods for parabolic distributed optimal control problems

We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrangian multiplier. Our final formulation fulfills the Babuška-Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a-posteriori error estimator which is efficient and reliable. Numerical experiments confirm our theoretical findings.

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arXiv2022

A DPG method for Reissner-Mindlin plates

We present a discontinuous Petrov-Galerkin (DPG) method with optimal test functions for the Reissner-Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

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arXiv2021

DPG methods for a fourth-order div problem

We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.

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arXiv2021

MINRES for second-order PDEs with singular data

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.

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arXiv2021

A DPG method for shallow shells

We develop and analyze a discontinuous Petrov--Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis--Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree only of the tangential displacement trace variable.

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arXiv2021

Multilevel decompositions and norms for negative order Sobolev spaces

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in $H^{-s}$ for $s\in (0,1)$. Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.

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arXiv2021

Analysis of backward Euler primal DPG methods

We analyse backward Euler time stepping schemes for the primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in the natural norm and in the $L^2$ norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.

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arXiv2020

A robust DPG method for large domains

We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincaré-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff--Love plate bending model, and also include fully discrete variants where optimal test functions are approximated. Numerical experiments for both model problems, including an-isotropic domains and mixed boundary conditions, confirm our findings.

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arXiv2020

A short note on plain convergence of adaptive least-squares finite element methods

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

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arXiv2020

A locking-free DPG scheme for Timoshenko beams

We develop a discontinuous Petrov-Galerkin scheme with optimal test functions (DPG method) for the Timoshenko beam bending model with various boundary conditions, combining clamped, supported, and free ends. Our scheme approximates the transverse deflection and bending moment. It converges quasi-optimally in $L_2$ and is locking free. In particular, it behaves well (converges quasi-optimally) in the limit case of the Euler-Bernoulli model. Several numerical results illustrate the performance of our method.

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arXiv2019

Space-time least-squares finite elements for parabolic equations

We present a space-time least squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space-time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space-time adaptivity. We also present a-priori error analysis on simplicial space-time meshes which are highly structured. Numerical results conclude this work.

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arXiv2019

Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation

We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov--Galerkin (DPG) methods. We investigate Fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration. Numerical experiments are presented in the case of uniform and adaptive mesh-refinement.

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arXiv2019

An ultraweak formulation of the Reissner-Mindlin plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation of the Reissner-Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness $t$. We also prove weak convergence of the Reissner-Mindlin solution to the solution of the corresponding Kirchhoff-Love model when $t\to 0$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory covers the case of non-convex polygonal plates. A numerical experiment for some smooth model solutions with fixed load confirms that our scheme is locking free.

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arXiv2019

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

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arXiv2019

Trace operators of the bi-Laplacian and applications

We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity, under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a non-closed space. This can be fixed by switching to the Kirchhoff-Love traces from [Führer, Heuer, Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp., 88 (2019)]. Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the DPG method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the non-closedness of a trace space), our analysis applies to any space dimension larger than or equal to two.

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arXiv2019

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

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arXiv2018

Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two

We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.

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